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328                     ANALYTICAL MECHANICS
By Hooke's law the force which produces the extension of the string a harmonic force, that is, if Q denotes the force then Q = — kq, where i a constant. Therefore the potential energy of the system is
tf = - f
But Q = mg when q = — Z>.   Therefore nig = A:/), or k = ™j^ .   Mak this substitution in the expression for the potential energy we obtain
Therefore the total energy of the system is E = 7T + (/
Differentiating the lust equation with respect to / we* obtain
which is the equation of simple harmonic motion.   Therefore
inu         // i  / ——(t + t.
and                      P = 2
It will be observed that, as in the case of every true harmonic moti< the period is not affected by the amplitude.
When the mass of the spring in negligible compared with that, of 1 suspended weight the last two equations become
q = ami \J & (t <f/M),
Therefore in this case the length of th(v equivalent simple pendulum equ
the stretch in the length of the spring produced by suspending the weig
2. A particle of mass m is attached to the middle point, of a stretcl
•elastic string of natural length L, modulus of elasticity X, and of negligi